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1 VOL.6 R 2 APRIL 1951 Philips Research Reports': EDITED BY THE RESEARCH LABORATORY OF. V. PHILIPS' GLOEILAlIIPEFABRIEKE, EIDHOVE, ETHERLADS R 160 Philips Res. Rep. 6, 81-85, 1951 THE DISCHARGE OF A SERIES OF EQUAL CODESERS HAVIG ARBITRARY RESISTACES COECTED I PARALLEL by H. BREMMER : Summary This paper concerns the discharge of a series of condensers through a ballistic', galvanometer if a variable resistance is connected in parallel to each condeneer. The flow of current through the galvanometer is calculated with the aid of the operational calculus. Résumé eet article traite de la décharge par un galvanomètre balistique d'une série de condensateurs avec une résistance variable montée en parallèle sur chacun d'eux. On détermine le courant qui traverse Ie galvanomètre à l'aide du calcul symbolique. Zusammenfassung Dieser Artikel behandelt die Entladung einer Reihe Kondensatoren durch ein ballistisches Galvanometer, wenn ein veränderlicher Widerstand zu jedem Kondensator parallelgeschaltet ist. Der Stromdurchgang durch das Galvanometer wird mit Hilfe der Operatorenrechnung ermittelt. 1. Introduetion A recent paper by Van Geel and Scholte I) deals with the electrical properties of a layer of' aluminium oxide which was deposited on, an aluminium background by means of anodic oxidation. The behaviour of such a layer could be described with the aid of a model consisting of a series of increasing resistances RI' R 2,..., R' across each of which a condenser of capacity LIe is connected in parallel (see fig. 1). In. the experiments of Van Geel and Scholte measurements were made, amongst others, after charging the capacitive system first and then discharging it through the galvanometer. The dependence of this charge' on LIe and the resistances is discussed in this paper with the aid of the operational calculus. The method developed may be of some interest because it can also be applied to other, similar, problems.

2 82 H.BREMMER 2. Derivation of the differential equations of the network We introduce the following symbols: in and in fot the currents through Rn and the condenser parallel to it, respectively, io for the current through the terminals at A and B, Ro for the internal resistance of the ballistic galvanometer, V for the d.c. voltage used for charging the system. Moreover, we indicate by t = 0 the moment of switching off the voltage V. The current io at the terminals then flows through the voltage source (supposed to have a negligible internal resistance) for t <0, and through the galvanometer for t>o. 7i h 73 R Fig. 1. The network considered in thi~ paper. The continuity of charge at the junctions hetwèen the nth and the (n + 1)th circuit requires in + in = in+! +in+!, while, moreover, io = i 1 +il' We therefore obtain: (n = 1,2,..., ). (1) FUrther, the potential difference across the nth condenser being given by Rnin, we get the following current tluough this condenser:. din Ln = LlCRn- dt (n = 1, 2,, ) (2) An additional relation for io has to he considered separately. The difference of the potentials at B and A amounts to V for t<o, and to -ioro for t >0. This petenrial difference can be interpreted, at any time, as the sum of the following two contributions: (1). the amount VU(-t),.arising from the electromotive -force applied at t<o. The unit function U(x) is defined here by 1 for x>o and by o for x<o;

3 DISCHARGE OF A SERIES OF CODESERS ----_._---_. 83 (2) the amount - Roio * if io* represents the difference of the actual current io and the direct current produced by V, viz. V io* = io U(-t). ~s; n~l The-first contribution vanishes for t>o, the second contribution for t<o. The potential difference between B and A also equals ~ Rnin; heuce: ~ Rnin = VU(-t) - Roio *. n=l ~ The set of the (2 + 1) equations (1), (2) and (4) suffices for a determination of the unknown functions il,i2,... ; i' il' i2'... -i» and io, if io has been replaced by io* according to (3). 3. Operational determination of the current io With a view to the calculation of the charge Q flowing through the galvanometer, we are particularly interested in the c1?'rent io. We first eliminate the functions il'...,i» by differentiating (1) and (4) while applying (2) and (3)2). With regard to the differentiation of (3) and (4) we recall the relation d d~ U(-x) = -<5(-x) = -<5(x), 1 (3) (4) <5(x)being Dirac's impulse function. The evaluation of the differentiations of (1) and (4) then finally leads to din in dio* V -+--=----<5(t) dt LlCRn dt s; ' n=l 1. dio* - ~ Ln = -Ro--- V<5(t)..dC n=l dt (n = 1, 2,..., ) (5) The functions i l,..., t» can be eliminated with the aid of-the operational method. The functions io*, i l,..., i all' vanish for t < 0 as well as for t-* + 00 and therefore have operational images for Re p >0 according to co io*(t).:!;= go(p) = p fe-f' io*(t) dt, - -co co in(t) ~~ gn(p) = p fe-pi in(t) dt. -ce (n'= 1,2,..., ) (6)

4 84 11.DREMMER The operational transposition of (5) can be performed by an application of the- differentiation rule, viz. and of the operational The result becomes relation b(t) '. p. (p + LI~RJ gn = pgo --:-p V, (n = 1,2,..., ) ~ Rn n=1 1 - ~ gn= LlC n=1 -Ropgo-pV. A substitution of the values of gl'..., g according to the former equation into the latter yields a relation containing go only. The final result for go then proves to be 1 ~ -~ n; n=il1cp + I/Rn n=1 1 :s +Ro n=il1cp + I/Rn 1 in which ~ represents the total impedance between A and B n=il1cp+l/rn for the frequency (J) = p/i. The charge flowing through the ballistic galvanometer, viz. co Q = I io*(t) dt, -co can be computed at once from the formula Q = lim go(p), p~o p as follows from (6). We obtain ~ Rn 2 - Q = V LIC,,=1. ~. Rn (Ro + ~ Rn) n=l n==l The elementary formula for a single resistance R parallel to the condeuser LIC corresponds to the special-case = 1, yielding R -Q= V LlC. Ro+R (7)

5 DISCHARGE OF A SERIES OF CODESERS 85 Formula (7) gives also information about the beginning of the discharge if we develop the p-function into a series of p-l. According to Heaviside's first expansion theorem the corresponding series of io is obtained by replacing each term p-b by t"in!. We get Vc (I I) io*(t)=-->l t+ Ro ( LlC n; ~ s; I + 2(Ll C)2 (~o+ ~IR) (:: +~l;jt ~, which series describes the gradual decreasing of io* after the jump -- VIRo at t = o. The duration of appreciable values of io* should be short compared to the vibration period of the ballistic galvanometer in order to get a reliable measurement of Q. Eindhoven, September 1950 n=1 REFERECES 1) W. Ch. van Geel and J. W. A. Scholte, Capacitë et pertes diëlectriques d'une couche d'oxyde dëposeë par oxydation anodique sur de l'aluminium, Philips Res. Rep. 6, 54-74, ) The method applied in this section is discussedin Chapters VIn and IX of Bal th. van der Pol and H. Bremmer, "Operational calculus based on the two-sided Laplace integral", Cambridge, University Press, 1950.

Chapter 28. Direct Current Circuits

Chapter 28. Direct Current Circuits Chapter 28 Direct Current Circuits Circuit Analysis Simple electric circuits may contain batteries, resistors, and capacitors in various combinations. For some circuits, analysis may consist of combining